Novi Sad J. Math.
Vol. 45, No. 1, 2015, 241-251
A MASSERA TYPE THEOREM IN
HYPERFUNCTIONS IN THE REFLEXIVE LOCALLY
CONVEX VALUED CASE
Yasunori Okada12
Abstract. We continue our study on Massera type theorems in hyperfunctions from [11] and [12]. In the latter, we gave a result in hyperfunctions with values in a reflexive Banach space. In this article, we report
its generalization to the case of hyperfunctions with values in a reflexive
locally convex space.
AMS Mathematics Subject Classification (2010): Primary 32A45; Secondary 32K13
Key words and phrases: bounded hyperfunctions; Massera type theorems
1.
Introduction
In [9], Massera studied the existence of a periodic solution to a periodic
ordinary differential equation, and gave the result that for a periodic linear ordinary differential equation of normal form, the existence of a bounded solution
in the future implies the existence of a periodic solution.
Theorem 1.1 ([9, Theorem 4]). Consider an equation
dx
= A(t)x + f (t),
dt
where A : R → Rm×m and f : R → Rm are 1-periodic and continuous. Then,
the existence of a bounded solution in the future (i.e., a solution defined and
bounded on a set {t > t0 } with some t0 ) implies the existence of a 1-periodic
solution.
Note that the inverse implication follows from the boundedness of a periodic
C 1 -function and therefore we have the equivalence between the existence of a
bounded solution in the future and that of a 1-periodic solution.
There appeared many generalizations of Theorem 1.1, and there arose a
question whether such phenomena appear commonly in periodic linear equations. For example, we refer to Chow-Hale [1] and Hino-Murakami [3] for functional differential equations with finite or infinite delay, to Shin-Naito [16] and
Naito-Nguyen-Miyazaki-Shin [10] for Banach valued cases, and to Zubelevich
1 Department of Mathematics and Informatics, Graduate School of Science, Chiba University, Chiba, 263-8522, Japan, e-mail:[email protected]
2 Supported by JSPS KAKENHI Grant Numbers 22540173, 23540186.
242
Yasunori Okada
[17] for discrete dynamical systems in reflexive Banach spaces and those in sequentially complete locally convex spaces with the sequential Montel property.
See also the references therein.
Being interested in these results, we have studied such phenomena in the
framework of hyperfunctions, and gave a Massera type theorem for hyperfunctions in [11] and its reflexive Banach valued variant in [12]. In this article, we
report a generalization of the latter result to the case of hyperfunctions with
values in a reflexive locally convex space.
The plan of this paper is as follows. In the section 2, we prepare some
notions and related terminologies on bounded hyperfunctions at infinity and
operators of type K introduced in [11, 12], and give our main result Theorem 2.3. In the section 3, we recall and study duality and compactness results
of the spaces of holomorphic functions taking values in a reflexive locally convex
space. In the last section 4, we give the proof of our main result.
2.
Main result
In this section, we recall some notions and terminologies briefly and state
our main result, Theorem 2.3. As for the preparation part, we follow [12, §2]
and refer to [11, §2 and §3] for details. See Sato [13, 14], Kawai [5], SatoKawai-Kashiwara [15], and Kaneko [4], for original hyperfunctions, Fourier
hyperfunctions, and related topics.
2.1.
Bounded hyperfunctions and classes of operators
Let us first recall the notion of bounded hyperfunctions at infinity.
We take a compactification D1 := R t {±∞} of R, and by considering the
diagram:
C = R + iR
⊂
D1 + iR
∪
∪
R = ]−∞, +∞[ ⊂ D1 = [−∞, +∞]
we identify C with an open subset of D1 + iR.
Let E be a sequentially complete Hausdorff locally convex space. We denote
by N (E) the family of continuous semi-norms of E, and by EO the sheaf of
E-valued holomorphic functions on C.
Definition 2.1. (1) The sheaf EOL∞ of E-valued bounded holomorphic functions at infinity on D1 + iR is defined by
OL∞ (U ) = {f ∈ EO(U ∩ C) | ∀L b U, f is bounded on L ∩ C}
E
for any open set U ⊂ D1 + iR.
(2) The sheaf BL∞ of E-valued bounded hyperfunctions at infinity on D1 is
defined as the sheaf associated with the presheaf
OL∞ (U \ Ω)
,
EO ∞ (U )
L
E
D1 ⊃ Ω 7→ lim
−→
U
A Massera type theorem in hyperfunctions....
243
for any open set Ω ⊂ D1 . Here U runs through complex neighborhoods
of Ω, that is, open sets in D1 + iR, including Ω as a closed subset.
The space EOL∞ (U ) is endowed with a natural locally convex topology by
the family of semi-norms
f 7→ sup p(f (w)),
w∈L∩C
where L runs through compact subsets in U and p runs through continuous
semi-norms of E. In the scalar case (E = C), we use abbreviations OL∞ and
BL∞ instead of COL∞ and CBL∞ , respectively. We also use the abbreviation
E
B of EBL∞ |R .
We list up some properties of bounded hyperfunctions. Refer to [11, §2] for
the precise statements and the proofs.
• BL∞ is an extension of B to D1 . That is, BL∞ |R = B.
• BL∞ is flabby. (In general, vector valued variants are not.)
• A section in BL∞ (]a, +∞]) admits a boundary value representation.
• There exists a natural embedding L∞ (]a, +∞[) ,→ BL∞ (]a, +∞]).
• The space BL∞ (D1 ) of the global sections of our sheaf BL∞ (in scalar
valued case) can be identified with the space BL∞ of bounded hyperfunctions (in 1-dimensional case) due to Chung-Kim-Lee [2].
Let us next recall a class of operators acting on bounded hyperfunctions at
infinity.
Let K = [a, b] be a closed interval in R (including the case K = {a}), and
U an open set in D1 + iR. Consider a family P = {PV }V ⊂U for open subsets
V ⊂ U of continuous linear maps
PV : EOL∞ (V + K) → EOL∞ (V ).
Note that the vectorial sum V + K := {w + t | w ∈ V, t ∈ K} is well-defined
even in case V 6⊂ C under the convention w + t = w for w = ±∞ + is ∈ V \ C
and t ∈ K.
Definition 2.2 (Operators of type K). P is said to be an operator of type
K for EOL∞ on U , if the diagram below commutes for any pair of open sets
V1 ⊃ V2 in U .
OL∞ (V1 + K)
E
PV1
OL∞ (V1 )
E
restriction
OL∞ (V2 + K)
E
restriction
PV2
OL∞ (V2 )
E
An operator P of type K for EOL∞ on U induces a family of linear maps
PΩ : EBL∞ (Ω + K) → EBL∞ (Ω),
for open sets Ω ⊂ D1 ∩ U ,
commuting with restrictions. An operator of type K = {0} corresponds to a
local operator, while an operator of type K = [−r, 0] corresponds to an operator
of finite delay r.
244
Yasunori Okada
2.2.
Massera type theorem in the reflexive valued case
Let us also recall the terminologies of EOL∞ -solutions and EBL∞ -solutions
to equations, and the notion of ω-periodicity for bounded hyperfunctions and
for operators of type K.
Let P be an operator of type K = [a, b] ⊂ R for EOL∞ on U . For an
open set V ⊂ U and a section f ∈ EOL∞ (V ), we say that u is an EOL∞ solution to the equation P u = f on V , or an EOL∞ (V )-solution to P u = f ,
if u belongs to EOL∞ (V + K) and satisfies PV u = f . Similarly, for an open
set Ω ⊂ D1 ∩ U and f ∈ EBL∞ (Ω), an EBL∞ -solution to P u = f on Ω is a
section u ∈ EBL∞ (Ω + K) satisfying PΩ u = f . Moreover, when f is a germ
of EBL∞ at +∞ (that is, f ∈ (EBL∞ )+∞ ), it makes sense to consider an
(EBL∞ )+∞ -solution to the equation P u = f .
The ω-translation operator Tω : u 7→ u(·+ω) for a positive constant ω, is an
operator of type {ω}, and the ω-difference operator Tω −1 is an operator of type
[0, ω]. A section u ∈ EBL∞ (Ω + [0, ω]) is called ω-periodic if it is an EBL∞ (Ω)solution to the equation (Tω − 1)u = 0, and an operator P of type K is called
ω-periodic if PV ◦Tω = Tω ◦PV +ω holds as maps EOL∞ (V +K +ω) → EOL∞ (V )
for any V .
Then, we have,
• Every ω-periodic hyperfunction f ∈ EB(R) has the unique ω-periodic
extension fˆ ∈ EBL∞ (D1 ).
• Every ω-periodic bounded hyperfunction f ∈ EBL∞ (D1 ) admits an ωperiodic boundary value representation.
• An ω-periodic operator of type K preserves the ω-periodicity of its operands.
Now we can state our main result. Let P be an ω-periodic operator of type
K = [a, b] for EOL∞ on a strip neighborhood D1 + i]−d, d[ of D1 with some
d > 0, and f ∈ EB(R) an ω-periodic E-valued hyperfunction. The unique
ω-periodic extension of f in EBL∞ (D1 ) is denoted also by the same symbol f
by the abuse of the notation.
Theorem 2.3. Assume that E is a reflexive locally convex space. Then,
P u = f has an ω-periodic EB(R)-solution if and only if it has an (EBL∞ )+∞ solution.
3.
Duality and compactness for EO(L)
Throughout this section, E denotes a reflexive locally convex space over C
and E 0 denotes its strong dual space. We recall the weak form of Köthe duality
for EO from [12, §3], and study a compactness result of EO.
A Massera type theorem in hyperfunctions....
3.1.
245
A weak form of the Köthe duality
Consider the space EO(L) := limV cL EO(V ) endowed with the locally con−→
vex inductive limit topology for a compact set L ⊂ C, where V runs through
the open neighborhoods of L in C. We give a weak form of the Köthe duality.
Let us cite two definitions ([12, Definition 3.1 and 3.2]).
Definition 3.1. For open neighborhoods V, W ⊂ C of L, we take a compact
neighborhood M of L in W ∩ V whose boundary γ := ∂M consists of finite
piecewise smooth simple closed curves, and define a bilinear form
0
h·, ·iL : E O(W \ L) × EO(V ) → C
by
Z
hF, f iL :=
(3.1)
F (w)(f (w))dw
γ
0
for F ∈ E O(W \ L) and f ∈ EO(V ). Here F (w)(f (w)) is a value of the
continuous linear functional F (w) ∈ E 0 evaluated at f (w) ∈ E.
Definition 3.2. Let L be a compact set in C and W an open neighborhood. We
0
0
define linear maps α : (EO(L))0 → E O(W \ L) and β : E O(W \ L) → (EO(L))0
by
1
1
(3.2)
α(ϕ)(w)(x) := ϕ
x ∈ C,
2πi w − ·
for ϕ ∈ (EO(L))0 , x ∈ E and w ∈ W \ L, and by
β(F )(f ) := hF, f iL ,
0
for F ∈ E O(W \ L) and f ∈ EO(L). Here we regard
E
O(L) in the right hand side of (3.2).
1
1
2πi w−· x
as an element of
Then, we can show the well-definedness of h·, ·iL , α, β, and the continuity of
α and β. Moreover, we can also show that β ◦ α = id(EO(L))0 and that the range
0
of α ◦ β − id(E0O(W \L)) is included in E O(W ). These facts give the following
results. (See Theorem 3.10 and Corollary 3.11 of [12].)
Theorem 3.3. Let E be a reflexive locally convex space. The maps α and β
induce the isomorphism between vector spaces
∼
0
0
(EO(L))0 −
→ E O(W \ L)/E O(W ).
Corollary 3.4 (Köthe duality). Let E be a reflexive locally convex space. The
maps α and β also induce the isomorphism between vector spaces
∼
0
(EO(L))0 −
→ E O ◦ (P1 \ L).
0
0
Here E O ◦ (P1 \ L) denotes the subspace {F ∈ E O(C \ L) | lim|w|→∞ F (w) = 0}
0
of E O(C \ L).
See Köthe [8, §27.3] for the classical Köthe duality.
246
3.2.
Yasunori Okada
Montel type lemma
Consider a compact set L ⊂ C and an open neighborhood V of L in C. As
0
before, h·, ·iL : E O ◦ (P1 \ L) × EO(V ) → C denotes the bilinear form given by
(3.1), used in the Köthe duality.
Lemma 3.5 (Montel type lemma). Let E be a reflexive locally convex space,
and {fn }n∈N a bounded sequence in EO(V ). Then, there exists f ∈ EO(V )
0
satisfying the following property: For any F ∈ E O ◦ (P1 \ L), we can take a
subsequence {n(k)}k such that
lim hF, fn(k) iL = hF, f iL .
k→∞
This lemma reflects the fact that bounded sets in EO(L) are precompact
with respect to the weak topology.
Note that when E is a reflexive Banach space, the subsequence {n(k)}k can
be taken independently of hF, ·iL . But we can not expect sequential precompactness in the general case. Since we want to use sequential convergence in
the proof of Theorem 2.3, we gave a statement in terms of sequences.
Proof of Lemma 3.5. Note that since E is reflexive, any bounded set in E is
weakly relatively compact. We denote by Ew the space E endowed with the
weak topology. Also note that bounded sets in EO(V ) are equicontinuous as
a family of maps from V to E, which follows from the Cauchy estimate. The
proof consists of three steps (I), (II) and (III).
(I) the choice of f and its holomorphy.
For any l ∈ N and any compact L ⊂ V , the set {fn (w) | n ≥ l, w ∈ L} in
E is bounded, and therefore
(3.3)
Al,L := the weak closure of {fn (w) | n ≥ l, w ∈ L}
is weakly compact
(i.e., compact in the topology induced from Ew ). We conQ
sider B := w∈V A0,{w} ⊂ (Ew )V endowed with the direct product topology,
that is, the topology of pointwise convergence. Then, B is compact (from
Tychonoff’s theorem). We also consider
(3.4)
Bl := the closure of {fn | n ≥ l} in (Ew )V
for l ∈ N. Then they are non-empty compact
subsets in B and decreasing in l,
T
and they share a common element f ∈ l Bl .
Each Bl is equicontinuous as a family of maps from V to Ew , since the
closure of an equicontinuous family with respect to the topology of pointwise convergence is also equicontinuous. (See, for example, Kelley-Namioka
[7, Chap.2, 8.12].) Moreover, it follows from Kelley [6, Chap.7, Theorem 15]
that the topology of pointwise convergence of the equicontinuous family Bl
coincides with its topology of convergence on compact sets. Therefore, f is a
uniform limit of some subnet of the sequence {fn }n consisting of Ew -valued
A Massera type theorem in hyperfunctions....
247
holomorphic functions on V , which implies that f itself is holomorphic as an
Ew -valued map on V . Since the E-valued holomorphy and the Ew -valued holomorphy are equivalent for E-valued maps, f is holomorphic as a map V → E,
that is, f ∈ EO(V ).
(II) the choice of {n(k)}k according to L and F .
For given L and F , we take a contour γ and its compact neighborhood
Γ ⊂ V \ L. The set A0,Γ (see (3.3) with L = Γ) is bounded in E as we have
seen in the part (I), and the correspondence
E 0 3 y 7→ q(y) := sup |y(x)|
x∈A0,Γ
defines a continuous semi-norm q on E 0 . Since {F (w)}w∈Γ is compact in E 0 ,
we have M := supw∈Γ q(F (w)) < +∞, which in particular implies
(3.5)
|F (w)(g(w))| ≤ M, for any g ∈ B0 and w ∈ Γ
with B0 defined in (3.4). Moreover, since w 7→ F (w)(g(w)) is holomorphic
as was shown in [12, Lemma 3.2], {w 7→ F (w)(g(w)) | g ∈ B0 } becomes an
equicontinuous family of functions on Int Γ.
Take a dense and countable subset C = {w1 , w2 , . . . } of γ, and define a
neighborhood Wk of the origin in Ew by
Wk := {x ∈ E | sup |F (wj )(x)| < 1/k}
1≤j≤k
for any k ≥ 1. Recall that f belongs to the closure of {fn }n≥l in (Ew )V with
respect to the topology of pointwise convergence, for any l ∈ N. Using again
the coincidence of the topology of pointwise convergence and the topology of
convergence on compact sets for an equicontinuous family, we can easily see
that the set
{fn }n≥l ∩ {g ∈ (Ew )V | ∀w ∈ γ, g(w) − f (w) ∈ Wk }
is non-empty. Therefore we can take n(k) for each k ≥ 1 satisfying n(1) <
n(2) < · · · and
∀w ∈ γ, fn(k) (w) − f (w) ∈ Wk .
(III) the convergence of hF, fn(k) iL to hF, f iL as k → ∞.
We define C-valued holomorphic functions hk and h on V \ L by
hk (w) := F (w)(fn(k) (w)),
h(w) := F (w)(f (w)),
and consider the sequence {hk }k≥1 . Since each fn belongs to B, it follows from
(3.5) that
|hk (w)| ≤ |F (w)(fn(k) (w))| ≤ M
248
Yasunori Okada
for w ∈ Γ and k ≥ 1. Therefore, as we have already seen, {hk }k is an equicontinuous family on holomorphic functions on Int Γ. Moreover, for wj ∈ C, it
follows from fn(k) (wj ) − f (wj ) ∈ Wk for k ≥ j that
|hk (wj ) − h(wj )| = |F (wj )(fn(k) (wj ) − f (wj ))| < 1/k.
This estimate shows that hk (wj ) → h(wj ) as k → ∞ for each wj ∈ C. In
other words, the sequence {hk }k converges to h with respect to the topology
of convergence on each point in C.
Note that on an equicontinuous family, the topology of convergence on each
point in a given subset coincides with the topology of convergence on each point
in the closure of the subset. (See Kelley-Namioka [7, Chap.2, 8.13].) Since C
is dense in γ, we have that hk (w) → h(w) as k → ∞ on each point in γ.
Now we can use Lebesgue’s bounded convergence theorem to show
Z
Z
hk (w)dw =
h(w)dw,
lim
k→∞
γ
γ
that is,
lim hF, fn(k) iL = hF, f iL ,
k→∞
which concludes the proof.
4.
Proof of the main theorem
Using the preparation in the section 3, we can prove our main theorem.
Proof of Theorem 2.3. The necessity follows from [12, Corollary 2.6], and we
shall prove the sufficiency.
Assume that P u = f has an (EBL∞ )+∞ -solution u. In a parallel manner
as in the proof of Theorem 4.4 of [12], we can take ũ ∈ EOL∞ (U̇ + K), f˜ ∈
E
OL∞ (D1 + iḂd ) satisfying (Tω − 1)f˜ = 0 and g ∈ EOL∞ (U ) for some a ∈ R
and d > 0, such that
[ũ] = u on Ω,
[f˜] = f on D1 ,
PU̇ ũ − g = f˜ on U̇ ,
under the notations
Ω := ]a, +∞],
U := ]a, +∞] + iBd ,
U̇ := ]a, +∞] + iḂd = U \ D1 .
Also in the same way, we define
Sk ũ :=
k−1
1X
Tjω ũ|U̇ +K ∈ EOL∞ (U̇ + K),
k j=0
Sk g :=
k−1
1X
Tjω g|U ∈ EOL∞ (U ),
k j=0
for k ≥ 1. Then, we have that
(4.1)
PU̇ Sk ũ − Sk g = f˜ on U̇ for any k ≥ 1,
A Massera type theorem in hyperfunctions....
249
and that {Sk ũ}k∈N ⊂ EOL∞ ((U̇ + K) ∩ C) and {Sk g}k∈N ⊂ EOL∞ (U ∩ C) are
bounded.
We consider each pair (Sk ũ, Sk g) as an E-valued holomorphic function on
the disjoint union V := ((U̇ + K) ∩ C) t (U ∩ C). (Although two open sets
(U̇ + K) ∩ C and U ∩ C are not disjoint, we can reduce the problem to the case
of two disjoint open sets by translation.) Applying Lemma 3.5 to the sequence
{(Sk ũ, Sk g)}k∈N , we can get a pair (v, h) ∈ EO((U̇ + K) ∩ C) × EO(U ∩ C)
satisfying the following property:
0
(C) for any L1 b (U̇ + K) ∩ C, L2 b U ∩ C, F1 ∈ E O ◦ (P1 \ L1 ), and
0
F2 ∈ E O ◦ (P1 \ L2 ), there exists a subsequence {k(l)}l such that
lim hF1 , Sk(l) ũiL1 = hF1 , viL1 ,
(4.2)
l→∞
lim hF2 , Sk(l) giL2 = hF2 , hiL1 .
l→∞
A priori v belongs to EO((U̇ + K) ∩ C) and h belongs to EO(U ∩ C). Now,
we want to show
(1) v ∈ EOL∞ (U̇ + K),
(2) h ∈ EOL∞ (U ),
(3) the equality PU̇ v − h = f˜ in EOL∞ (U̇ ), and
(4) the ω-periodicity of v.
For this purpose, it suffices to show (4) and
(5) the equality PU̇ ∩C v − h = f˜ in EO(U̇ ∩ C).
In fact, we can easily prove the implications (4) ⇒ (1); (1), (5) ⇒ (2); and (1),
(2), (5) ⇒ (3).
In order to show (5), we take an arbitrary compact set L b U̇ and an
0
arbitrary F ∈ E O ◦ (P1 \ L). Then, we have, from (4.1), that
hF, PU̇ Sk ũ − Sk giL = hF, f˜iL ,
which implies
hPL∗ (F ), Sk ũiL+K − hF, Sk giL = hF, f˜iL .
0
0
Here PL∗ : E O ◦ (P1 \ L) → E O ◦ (P1 \ (L + K)) is the abstract adjoint operator of
PL whose existence is guaranteed by Corollary 3.4. We can apply the property
(C) to the two terms on the left hand side in the case L1 := L + K, L2 = L,
F1 = PL∗ (F ), F2 = F . Then by taking the limit in (4.2), we have
hPL∗ (F ), viL+K − hF, hiL = hF, f˜iL ,
or
(4.3)
hF, PU̇ ∩C v − hiL = hF, f˜iL .
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Yasunori Okada
Since L and F are arbitrary, the equality (5) follows from (4.3). Here we used
Corollary 3.4 again.
In order to show (4), we also take an arbitrary compact set L b (U̇ +K)∩C
0
and an arbitrary F ∈ E O ◦ (P1 \ L). Using the equality
(Tω − 1)Sk ũ =
1
(Tkω − 1)ũ
k
and the boundedness of {(Tkω − 1)ũ}k , we have
lim hF, (Tω − 1)Sk ũiL → 0.
k→∞
Then, by taking the adjoint, by applying the property (C), and by taking the
adjoint again, we can successively show the following
lim h(Tω − 1)∗ (F ), Sk ũiL+[0,ω] → 0,
k→∞
h(Tω − 1)∗ (F ), viL+[0,ω] = 0,
(4.4)
hF, (Tω − 1)viL = 0.
Since L and F are arbitrary, (4.4) implies (4).
By virtue of the ω-periodicity, v has a unique ω-periodic extension in
E
OL∞ (D1 +iḂd ). Moreover, h has a unique ω-periodic extension in EOL∞ (D1 +
iBd ). In fact, since h = PU̇ v − f˜ is ω-periodic on U̇ , it is ω-periodic also in U ,
and can be extended.
Finally note that [v] ∈ EBL∞ (D1 ) gives an ω-periodic solution.
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Received by the editors January 31, 2015